Measure of Self-Affine Sets and Associated Densities

Abstract

Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of ${\mathbb{R}}^n$ with $0\in \mathcal{D}$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set-valued equation $BK=\bigcup _{d\in \mathcal{D}}(K+d)$. In the case where $\text{card}(\mathcal{D})=|\det B|,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu =\underset{s\to \infty }{\lim }\sum _{{\ell }_0,\dots ,{\ell }_{s-1}\in \mathcal{D}}{\delta }_{{\ell }_0+B{\ell }_1+\cdots +B^{s-1}{\ell }_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<|\det B|$ and $B$ is a similarity matrix, we relate the Hausdorff measure ${\mathcal{H}}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.

20 pages

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